Question

Evaluate each of the following integrals:

(i) $\int x^4\mathrm{d}x$

(ii) $\int x^{\frac{5}{4}}\mathrm{d}x$

(iii) $\int \frac{1}{x^5}\mathrm{d}x$

(iv) $\int \frac{1}{x^{3/2}}\mathrm{d}x$

(v) $\int 3^x\mathrm{d}x$

(vi) $\int \frac{1}{\sqrt[3]{x^2}}\mathrm{d}x$

(vii) $\int 3^{2{\log }_3}{}^x\mathrm{d}x$

(viii) $\int {\log }_{x}x\mathrm{d}x$

Answer 1

(i)
$$\begin{gathered} \int x^{4}dx \\ =\frac{x^{4+1}}{4+1}+C \\ =\frac{x^5}{5}+C \end{gathered}$$

(ii)
$$\begin{gathered} \int x^{\frac{5}{4}}dx \\ =\frac{x^{{\displaystyle \frac{5}{4}}+1}}{{\displaystyle \frac{5}{4}}+1}+C \\ =\frac{4}{9}{x}^{\frac{9}{4}}+C \end{gathered}$$

(iii)
$$\begin{gathered} \int x^{-5}dx \\ =\frac{x^{{\displaystyle -5+1}}}{{\displaystyle -5}+1}+C \\ =-\frac{1}{4}{x}^{-4}+C \\ =-\frac{1}{4x^4}+C \end{gathered}$$

(iv)
$$\begin{gathered} \int \frac{dx}{x^{3/2}} \\ =\int x^{-3/2}dx \\ =\left[\frac{x^{-{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{-3}{2}}+1}\right]+C \\ =\left[\frac{x^{-{\displaystyle \frac{1}{2}}}}{-{\displaystyle \frac{1}{2}}}\right]+C \\ =-\frac{2}{\sqrt{x}}+C \end{gathered}$$

(v)
$$\begin{gathered} \int 3^{x}dx \\ =\frac{3^x}{\ln 3}+C \end{gathered}$$

(vi)
$$\begin{gathered} \int \frac{dx}{\sqrt[3]{x^2}} \\ =\int \frac{dx}{x^{2/3}} \\ =\int x^{-2/3}dx \\ =\frac{x^{-{\displaystyle \frac{2}{3}}+1}}{-{\displaystyle \frac{2}{3}}+1}+C \\ =3x^{\frac{1}{3}}+C \end{gathered}$$

(vii)
$$\begin{gathered} \int 3^{2{\log }_{3^x}}dx \\ =\int 3^{{\log }_3{x}^2}dx \\ =\int x^{2}dx \\ =\frac{x^3}{3}+\mathrm{C} \end{gathered}$$

(viii)
$$\begin{gathered} \int {\log }_{x}xdx \\ =\int 1·dx \\ =x+C \end{gathered}$$